A lower bound on the growth exponent for loop-erased random walk in two dimensions
A Lower Bound on the Growth Exponent for Loop-Erased Random Walk in Two Dimensions
The growth exponent α for loop-erased or Laplacian random walk on the integer lattice is defined by saying that the expected time to reach the sphere of radius n is of order nα. We prove that in two dimensions, the growth exponent is strictly greater than one. The proof uses a known estimate on the third moment of the escape probability and an improvement on the discrete Beurling projection theorem.
A mathematical model of an In Vitro experiment to investigate endothelial cell migration.
A matrix with an application to the motion of an absorbing Markov chain. I.
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A note on a two-person zero-sum game stimulated by a Markov chain
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A recurrence theorem for square-integrable martingales
Let be a zero-mean martingale with canonical filtration and stochastically -bounded increments which means that a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let . It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions....